We calculate the effect of a spatially dependent effective mass (SPDEM) [adapted from R. N. Costa Filho et al. Phys. Rev. A., textbf{84} 050102 (2011)] on an electron and hole confined in a quantum well (QW). In the work of Costa Filho et al., the tr
anslation operator is modified to include an inverse character length scale, $gamma$, which defines the SPDEM. The introduction of $gamma$ means translations are no longer additive. In nonadditive space, we choose a `skewed Gaussian confinement potential defined by the replacement $xrightarrowgamma^{-1}ln(1+gamma x)$ in the usual Gaussian potential. Within the parabolic approximation $gamma$ is inversely related to the QW thickness and we obtain analytic solutions to our confinement Hamiltonian. Our calculation yields a reduced dispersion relation for the gap energy ($E_G$) as a function of QW thickness, $D$: $E_Gsim D^{-1}$, compared to the effective mass approximation: $E_Gsim D^{-2}$. Additionally, nonadditive space contracts the position space metric thus increasing the occupied momentum space and reducing the effective mass, in agreement the relation: $m_o^{*-1}propto frac{partial^2 E}{partial v{k}^2}$. The change in the effective mass is shown to be a function of the confinement potential via a point canonical transformation. Our calculation agrees with experimental measurements of $E_G$ for Si and Ge QWs.
A translation operator is introduced to describe the quantum dynamics of a position-dependent mass particle in a null or constant potential. From this operator, we obtain a generalized form of the momentum operator as well as a unique commutation rel
ation for $hat x$ and $hat p_gamma$. Such a formalism naturally leads to a Schrodinger-like equation that is reminiscent of wave equations typically used to model electrons with position-dependent (effective) masses propagating through abrupt interfaces in semiconductor heterostructures. The distinctive features of our approach is demonstrated through analytical solutions calculated for particles under null and constant potentials like infinite wells in one and two dimensions and potential barriers.
